A conditional proof of the ISP for quasinilpotent operators

Abstract

The invariant subspace problem (ISP) is a well known unsolved problem in funtional analysis. While many partial results are known, the general case for complex, infinite dimensional separable Hilbert spaces is still open. It has been shown that the problem can be reduced to the case of operators which are norm limits of nilpotents. One of the most important subcases is the one of quasinilpotent operators, for which the problem has been extensively studied for many years. In this paper, we will introduce a new conjecture (supported by a heuristic argument), and we will prove conditionally that every quasinilpotent operator has a nontrivial invariant subspace. We will conclude by posing an open problem which would have deep implications regarding the ISP.